#### Instanton operators and symmetry enhancement in 5D supersymmetric gauge theories

Prog. Theor. Exp. Phys.
Instanton operators and symmetry enhancement in 5D supersymmetric gauge theories
Yuji Tachikawa 0 1
0 Institute for the Physics and Mathematics of the Universe, University of Tokyo , Kashiwa, Chiba 277-8583 , Japan
1 Department of Physics, Faculty of Science, University of Tokyo , Bunkyo-ku, Tokyo 133-0033 , Japan
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supersymmetric gauge theories in five dimensions often exhibit less symmetry than the ultraviolet fixed points from which they flow. The fixed points might have larger flavor symmetry or they might even be secretly 6D theories on S1. Here we provide a simple criterion when such symmetry enhancement in the ultraviolet should occur, by a direct study of the fermionic zero modes around one-instanton operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index B10, B14, B16 In four dimensions and in lower dimensions, we often start from a Lagrangian gauge theory in the ultraviolet (UV) and study the behavior of the system in the infrared (IR). In five dimensions (5D), Lagrangian gauge theories are always non-renormalizable, and instead one needs to look for nontrivial ultraviolet superconformal field theories (SCFTs) from which they flow out. With supersymmetry, such a study is indeed possible, as demonstrated in Refs. [1-4], by combining field-theoretical analyses and embedding into string theory. There and in other works, it was found that the UV fixed point can have enhanced symmetry: the instanton number symmetry sometimes enhances to a non-Abelian flavor symmetry, and sometimes enhances to the Kaluza-Klein mode number of a 6D theory on S1. For example, the UV SCFT for N = 1 SU(2) theory with N f 7 flavors has E N f +1 flavor symmetry; with N f = 8 flavors, the UV fixed point is instead a 6D N = (1, 0) theory with E8 symmetry compactified on S1 [5]. For SU(N ) theory, the flavor symmetry enhancement only occurs for some specific choice of the number of the flavors and of the Chern-Simons levels. N = 2 theory with gauge group G in 5D, in contrast, comes from some 6D N = (2, 0) theory compactified on S1, as originally found in the context of string duality [6,7]. These results were soon extended to include more models, using webs of five-branes, in, e.g., Refs. [8-10]. More recently, various sophisticated techniques such as supersymmetric localizations, Nekrasov partition functions, and refined topological strings have been applied to the analysis of these 5D systems. The symmetry enhancement of the models mentioned above has been successfully confirmed by these methods, and even more diverse models are being explored; see, e.g., Refs. [11-27].
1. Introduction
These results are impressive, but the techniques used are rather unwieldy.1 In this paper, we
describe a simpler method to identify the symmetry enhancement of a given 5D gauge theory,
assuming that it has an ultraviolet completion either in 5D or 6D. Although heuristic, this method tells us
what the enhanced flavor symmetry will be and whether the ultraviolet completion is a 5D SCFT or
a 6D SCFT on S1.
We do this by identifying the supermultiplet of broken symmetry currents2 by studying instanton
operators that introduce non-zero instanton number on a small S4 surrounding a point.3 When the
instanton number is one, the structure of the moduli space and the zero modes is particularly simple,
allowing us to find the broken symmetry currents rather directly. The spirit of the analysis will be
close to the analysis of monopole operators of 3D supersymmetric gauge theories in Ref. [30], except
that we need to identify an operator in the IR that would come from a UV current operator, rather
than vice versa.
The rest of the paper is organized as follows. In Sect. 2, we collect the basic facts on the
supersymmetry and on instantons on S4 that we use. In Sect. 3, we analyze N = 1 SU(2) gauge theories
with N f 8 flavors and N = 2 SU(2) theory. The effect of the discrete theta angle will also
be discussed. In Sect. 4, we extend the discussion to N = 1 SU(N ) gauge theories with
fundamentals and ChernSimons terms, and to N = 2 SU(N ) theory. We use the results that will be
obtained up to this point in Sect. 5 to study symmetry enhancement in quiver gauge theories made
of SU gauge groups and bifundamentals. We assume that the effective number of flavors of each
SU(N ) node is 2N and that the ChernSimons levels are all zero. We conclude with a discussion
in Sect. 6.
2. Preliminaries
2.1. Supermultiplet of broken currents
with scaling dimensions 3, 3.5, 4, 4, respectively. Here a is the adjoint index of the flavor symmetry,
i = 1, 2 is the index of Sp(1)R , and is the spinor index of SO(5); the symplectic Majorana condition
is imposed on ia.
1 One of the trickiest aspects is the need to remove spurious contributions from the so-called U(1) parts in
the instanton counting method and from the parallel legs in the toric diagram in the topological string method.
2 The importance and the special property of the current supermultiplets in 5D SCFTs were emphasized in
Ref. [13]. The analysis presented here is strongly influenced by that paper.
3 Recently, a paper [28] appeared in which instanton operators of 5D N = 2 gauge theories were also studied.
There the emphasis was on the multi-point functions of instanton operators. In this paper we just consider a
single instanton operator, and the multiplet it forms under the flavor symmetry and the supersymmetry. In another
recent paper [29], a different type of instanton operator was studied, where an external Sp(1)R background
was introduced on S4 to have manifest supersymmetry at the classical level. Here we do not introduce such
additional backgrounds.
4 For the details of the superconformal multiplets in 5D and in 6D, see, e.g., Ref. [31] and references therein.
The mass deformation is done by adding L = ha Ma to the Lagrangian, thus breaking some of
the flavor symmetry:
where (i j) is an Sp(1)R triplet scalar (i j), i is a symplectic Majorana Sp(1)R doublet fermion
i, and J is a vector. The component M in (2.1) is a divergence of J, and therefore is a descendant.
Suppose instead that the 5D theory is obtained by putting an N = (1, 0) 6D SCFT on S1, possibly
with a nontrivial holonomy for the flavor symmetry; we consider N = (2, 0) theories as special cases
of N = (1, 0) theories. The conserved current supermultiplet in 6D contains
where A is the 6D vector index and we have omitted the adjoint index for brevity. On S1, we have
KaluzaKlein (KK) modes J(n) and M(n) := J6(n) where n is the KK mode number. Then the 6D
conservation gives
2.2. Instanton operators
Suppose now that we are given a 5D gauge theory, and further assume that it is a mass deformation
of a 5D SCFT or a 6D SCFT on S1. Pick a point p in R5 and insert there a current operator that is
broken by the mass deformation in the former case, and a current operator with non-zero KK mode
number in the latter case. Surround p by an S4, on which we have a certain state. Let the size of S4
be sufficiently larger than the characteristic length scale of the system set by the mass deformation
or the inverse radius of S1. Then the state on S4 can be analyzed using the gauge theory. When the
instanton number of the gauge configuration on S4 is non-zero, we call the original operator inserted
at p an instanton operator.
It should be possible to study the supermultiplet structure of instanton operators in detail using
supersymmetric 5D Lagrangians on S4 times R or S1, using the results in, e.g., Refs. [11,3234].
Here we only provide a rather impressionistic analysis of one-instanton operators, i.e., the instanton
operators when the instanton number on S4 is one.
In the rest of this section we gather known facts on one-instanton moduli spaces and fermion
zero modes. We will be brief; a comprehensive account of SU(N ) instantons can be found in, e.g.,
Ref. [35]. For instantons of general gauge groups, see, e.g., Refs. [36,37].
When the gauge group is SU(2). Any instanton configuration on S4 can be obtained by conformal
transformations from one on a flat R4. When the gauge group is SU(2), the one-instanton moduli
space on R4 has the form
where R4 parametrizes the position, R>0 the size, and S3/Z2 the gauge rotation at infinity. When
the instanton is mapped to S4, the first two factors R4 and R>0 combine to form the ball B5 with a
standard hyperbolic metric.5 The asymptotic infinity of R4 is mapped to a point on S4, and therefore
the gauge symmetry there should really be gauged. Therefore we lose the last factor S3/Z2, but we
should remember that the gauge group SU(2) is broken to Z2.
A point in B5 very close to a point x in S4 describes an almost point-like instanton localized at x .
A point at the center of B5 corresponds to the largest possible instanton configuration, which is in
fact SO(5) invariant. This invariant configuration can be identified with the positive-chirality spinor
bundle on the round S4. This configuration is also known as Yangs monopole [42].
A Weyl fermion of the correct chirality in the doublet of the SU(2) gauge group has just
one zero mode on a one-instanton configuration. On S4, this is a singlet of SO(5) rotational
symmetry.
A Weyl fermion of the correct chirality in the adjoint of the SU(2) gauge group has four zero modes.
Two are obtained by applying supertranslations and the other two by applying special superconformal
transformations to the original bosonic configuration. When conformally mapped to S4, these four
modes transform in the spinor representation of SO(5). They are exactly the modes obtained by
applying the 5D supersymmetry to the bosonic instanton configuration. In this sense the instanton
configuration breaks all the supersymmetry classically, but this does not mean that the instanton
operator is in a generic, long multiplet of supersymmetry, as we will soon see.
When the gauge group is general. When the gauge group is a general simple group G, any
one-instanton configuration is obtained by embedding an SU(2) one-instanton configuration by a
homomorphism : SU(2) G determined by a long root. Therefore, the one-instanton moduli
space on R4 is of the form
where H is the part of G that is unbroken by the SU(2) embedded by . A further quotient
G/(SU(2) H ) is known as the Wolf space of type G. The form of H is well known; here we
only note that for G = SU(N ) we have H = U(1) SU(N 2).
The conformal transformation to S4 again combines R4 and R>0 to the ball B5, and we lose G/H
as before. The remaining effect is that we have H as the unbroken gauge symmetry. Analyzing the
fermionic zero modes of an arbitrary representation R of G around this configuration is not any
harder than for SU(2), since the actual gauge configuration is still essentially that of SU(2). We only
have to decompose R under SU(2) H , and to utilize our knowledge for SU(2).
5 The hyperbolic ball B5 is also known as the Euclidean AdS5. This fact was used effectively in a series of
early works relating AdS/CFT and instantons; see, e.g., Refs. [3841].
3. SU(2)
3.1. Pure N = 1 theory
After these preparations, let us first consider one-instanton operators of pure N = 1 SU(2) gauge
theory. As recalled in the previous section, the one-instanton moduli space on S4 is just the ball B5.
We consider a state corresponding to the lowest SO(5)-invariant wavefunction.
Now we need to take fermionic zero modes into account. The gaugino is a spinor field in five
dimensions, which gives two Weyl fermions with the correct chirality in the adjoint of the SU(2)
gauge group and in the doublet of Sp(1)R when restricted on S4. As recalled in the previous section,
the zero modes are in the doublet of Sp(1)R and in the spinor of SO(5) rotational symmetry. Let us
denote them by i, where i = 1, 2 is for Sp(1)R and = 1, 2, 3, 4 is for SO(5). The symplectic
Majorana condition in 5D means that we need to quantize these zero modes into gamma matrices
satisfying
The states on which these zero modes act, then, can be found by decomposing the Dirac spinor
representation of SO(8) in terms of its subgroup Sp(1)R SO(5) such that the vector representation
of SO(8) becomes the doublet times the quartet. We find the following sixteen states:
which form exactly the broken current supermultiplet (2.3) recalled in the last section. We put the
plus signs as superscripts to remind us that they are one-instanton operators.
Let us assume that this gauge theory is a mass deformation of a UV 5D SCFT. Then the UV SCFT
should simultaneously have both the multiplet that contains J+ in the gauge theory and the multiplet
that becomes the instanton number current
Now, J+ has charge 1 under J0, because J+ is a one-instanton operator. Therefore, they should form
an SU(2) flavor symmetry current. This conclusion agrees with the original stringy analysis [1].
Effect of the discrete theta angle. In the analysis so far, we have neglected the effect of the discrete
theta angle of the SU(2) theory. The theta angle is associated with 4(SU(2)) = Z2, and therefore
only takes the values = 0 or . On a five-manifold of the form M S1 such that the SU(2)
configuration on M has instanton number 1 and there is a nontrivial Z2 holonomy around S1, the theta
angle = assigns an additional sign factor 1 in the path integral.6 On R4 S1, this has an effect
that makes the wavefunctions on the one-instanton moduli space sections of a nontrivial line bundle
with holonomy 1 on S3/Z2.
In our setup, the supermultiplet (3.2) is kept when = 0, but is projected out when = .
Therefore, there is an enhancement of the instanton number symmetry to SU(2) when = 0, but we see no
enhancement when = . This effect of the discrete theta angle matches what was found originally
in Ref. [2] using stringy analysis.
A caveat here is that, in our crude analysis, we can only say that the states considered so far do
not give any broken current supermultiplet. It is logically possible that exciting non-zero modes in
the one-instanton sector or considering operators with instanton number 2 or larger gives rise to a
6 For a derivation, see, e.g., Appendix A of Ref. [43].
broken current supermultiplet, enhancing the instanton number symmetry to SU(2) even with = .
A careful study on this point is definitely worthwhile, but is outside the scope of this paper. The same
caveat is also applicable to the rest of the article, but we will not repeat it.
3.2. With fundamental flavors
Next, let us consider N = 1 SU(2) theory with N f flavors in the doublet. Stated differently, we add
2N f half-hypermultiplets in the doublet of SU(2). At the classical Lagrangian level, they transform
under SO(2N f ) symmetry.
In a one-instanton background, they give 2N f fermionic zero modes, which need to be quantized
as gamma matrices:
{
where a = 1, . . . , 2N f is the index of the vector representation of SO(2N f ). They act on the Dirac
spinor representation S+ S of SO(2N f ). Tensoring with the result (3.2) of the quantization of
the adjoint zero modes, we find that the one-instanton operator is a broken current supermultiplet in
the Dirac spinor of SO(2N f ).
Now we need to take the unbroken Z2 gauge symmetry into account. The generator of this Z2
symmetry acts by 1 on the doublet representation, and therefore by 1 on the gamma matrices
in (3.4). Therefore, this Z2 acts as the chirality operator on S+ S. Therefore, depending on the
value of the discrete theta angle being 0 or , we keep only one-instanton operators in S+ or S.
These two choices are related by the parity operation of flavor O(2N f ) and are therefore equivalent.
In conclusion, we found the following current multiplets:
the conserved SO(2N f ) currents J[ab],
the conserved instanton number current J0,
the broken currents coming from one-instanton operators J+,A where A is the chiral spinor
index of SO(2N f ).
Take 1 N f 7. If we assume that the gauge theory is a mass deformation of a UV fixed point,
we see that the currents listed above need to combine to give the flavor symmetry E N f +1. This can
be seen by attaching an additional node, representing a Cartan element for J0, to the node of the
Dynkin diagram of SO(2N f ) that gives the chiral spinor representation. For example, we have
when N f = 7, where the black node is for the instanton number current.7 This enhancement pattern
agrees with what was found originally in Ref. [1].
Let us boldly take N f = 8. We now have the combined Dynkin diagram
which is known as E9 or E8. If we assume that the gauge theory is an outcome of a massive
deformation of some UV completion, the UV completion needs to have an affine E8 flavor symmetry as
7 Strictly speaking, this procedure is not unique when a theory with very small N f is considered alone. As a
simpler example of this issue, suppose we know that we have an SU(2) current, an instanton number current,
and a broken current coming from one-instanton operators in the doublet of SU(2). Then we have two choices,
either an SU(3) corresponding to or an Sp(2) corresponding to . They can only be distinguished
by studying the two-instanton operators. In our case, however, we can just study the N f = 7 case and then
apply the mass deformation to make N f smaller, to conclude that the flavor symmetry is always EN f +1.
a 5D theory. Stated differently, this means that the UV completion needs to be a 6D theory with E8
flavor symmetry. This conclusion matches what was found in Ref. [5].
We see a problem when N f > 8: the combined Dynkin diagram defines a hyperbolic KacMoody
algebra, whose number of roots grows exponentially. It is therefore unlikely that there is a UV
completion when N f > 8. Again, this matches the outcome of a different analysis in Ref. [1].
Before proceeding, we note here that it is well known that the instanton particles in this gauge
theory give rise to spinors of SO(2N f ) flavor symmetry. The only slightly new point in this section
is that, when studied in the context of instanton operators, they are indeed part of the broken current
multiplets given by (3.2).
3.3. N = 2 theory
As a final example of SU(2) gauge theory, let us consider N = 2 gauge theory. On the one-instanton
background on S4, we have four Weyl fermions in the adjoint of SU(2), and the zero modes can be
denoted by i where i = 1, 2, 3, 4 is now for Sp(2)R and = 1, 2, 3, 4 is for the spacetime SO(5).
Again, the symplectic Majorana condition in 5D means that they become gamma matrices with
the commutation relation
These gamma matrices act on the following one-instanton states on S4:
where a = 1, 2, 3, 4, 5 is the vector index for Sp(2)R = SO(5)R. The gamma-tracelessness condition
on +, Q+ and the tracelessness condition on +, T + need to be imposed.
When the discrete theta angle is zero, these operators are all kept, and the structure of the operators
is exactly that of KK modes of the 6D N = 2 energymomentum supermultiplet. For example, the
currents J[+ab] in the adjoint of Sp(2)R suggest that the Sp(2)R symmetry of the 5D gauge theory
enhances to the affine Sp(2)R, and the symmetric traceless T(+) is the KK mode of the 6D energy
momentum tensor. This is as it should be, since the S1 compactification of N = (2, 0) theory of type
SU(2) on S1 is described by N = 2 SU(2) gauge theory in 5D. The relation between the 5D N = 2
theory and the 6D N = (2, 0) theory on S1 has been extensively studied; see, e.g., Refs. [44,45].
When the discrete theta angle is , these operators are all projected out. In this case too, the gauge
theory is the IR description of N = (2, 0) theory of type SU(3) on S1 with an outer-automorphism
Z2 twist around it [43]. We expect operators with the same structure to arise in a sector with higher
instanton number, but to check this is outside the scope of this paper.
4. SU(N )
4.1. Pure N = 1 theory
Now let us move on to SU(N ) gauge theories. Our first example is the pure SU(N ) theory. As recalled
in Sect. 2, in one-instanton configurations on S4, the gauge fields take values in SU(2) SU(N ).
The unbroken subgroup is U(1) SU(N 2). We take the generator of the U(1) part to be
diag(N 2, N 2, 2, 2, . . . , 2).
In this normalization, when the ChernSimons level is , the one-instanton configuration has the
U(1) charge (N 2).
An adjoint Weyl fermion of SU(N ) on this background decomposes into an SU(2) adjoint Weyl
fermion, N 2 Weyl fermions in the doublet of SU(2) in the fundamental of SU(N 2) and with
U(1) charge N , and Weyl fermions that are neutral. The gaugino in 5D gives two adjoint Weyl
fermions of SU(N ) on S4. The SU(2) adjoint part gives the same broken current supermultiplet
(3.2), and we need to take additional 2 (N 2) zero modes coming from the SU(2) doublet into
account.
By quantizing them, we have fermionic creation operators Bia where i = 1, 2 is now for Sp(1)R
and a = 1, . . . , (N 2) is for SU(N 2). The U(1) charge of Bia is N . The SU(N 2) invariant
states are then
a1aN2 Bi1a1 Bi2a2 BiN2aN2 |0 ,
(Bia)2(N 2) |0 ,
with U(1) charge (N 2)N , 0, +(N 2)N , respectively.
When is neither 0 nor N , all states are projected out, due to the non-zero U(1) gauge charge.
When is N , one singlet state is kept, and we have a broken current supermultiplet (3.2). Therefore
we expect the enhancement of the instanton number symmetry to SU(2). This enhancement was
recently discussed in Ref. [19].
When is 0, the one-instanton operators are the tensor product of the broken current
supermultiplet (3.2) times a1aN2 Bi1a1 Bi2a2 BiN2aN2 |0 , which transforms in the N 1-dimensional
irreducible representation of Sp(1)R. This is a short supermultiplet, but does not correspond to a
broken flavor symmetry.
4.2. With fundamental flavors
Our next example is SU(N ) theory with N f hypermultiplets in the fundamental representation. On
one-instanton configurations on S4, each flavor decomposes into a pair of SU(2) doublets and a
number of neutrals; they all have U(1) charge N 2. Therefore, we have additional fermionic creation
operators Cs , s = 1, . . . , N f , of U(1) charge N 2. They act on the states of the form
for k = 0, . . . , N f , with U(1) charge (N 2)(k N f /2).
Tensoring (3.2), (4.2), and (4.3) and imposing the U(1) gauge neutrality condition, we see that a
broken symmetry supermultiplet survives when we have
Now, in Ref. [4] it was shown that we need || N N f /2 to have a 5D UV SCFT behind the
gauge theory. Therefore | N | N f /2. We also trivially have |k N f /2| N f /2. Therefore,
the equality (4.4) can only be satisfied when = N N f /2.
When = N N f /2, the surviving broken current supermultiplet comes from k = 0 or
k = N f in (4.3). They have instanton number one and baryonic charge N f /2. Therefore, when
= N N f /2, one U(1) enhances to SU(2), and, when = N N f /2, another U(1) enhances
to SU(2). When = N N f /2 = 0, the combination I B/N of the instanton charge I and the
baryonic charge B both enhance to SU(2), making the UV flavor symmetry SU(N f ) SU(2)+
SU(2). This enhancement pattern was found in Refs. [19,27].
4.3. N = 2 theory
Let us next consider N = 2 SU(N ) theory. The ChernSimons level is automatically zero. Again,
the adjoint Weyl fermions of SU(N ) decompose into those that are adjoint of SU(2) and those that
are doublets of SU(2). The zero modes of the former generate the states (3.8).
The zero modes of the latter give us fermionic creation operators Bia, where i = 1, 2, 3, 4 is for
Sp(2)R and a = 1, . . . , (N 2) is for gauge SU(N 2). The U(1) SU(N 2) neutral states then
have the form
a1aN2 Bi1a1 Bi2a2 BiN2aN2 b1bN2 B j1b1 B j2b2 B jN2bN2 |0
= Bi1 1 B j1 1 Bi2 2 B j2 2 BiN2 N 2 B jN2 N 2 |0 .
We want to decompose them under the action of Sp(4)R. Let us think that SU(4) acts on the indices
i and j . The indices in and jn are antisymmetrized. Combined, [in jn] is a vector of SO(6) SU(4).
The indices are symmetrized under the combined exchange of [in jn] and [im jm ]. Therefore, the states
(4.5) transform under the (N 2)nd symmetric power of the vector of SO(6). Decomposing it under
Sp(4)R SO(5)R SO(6), we see that the states (4.5) are in the representation
V0 V1 VN 2,
where Vk is the kth symmetric traceless representation of SO(5)R.
This structure is precisely what we would expect for the KK modes of N = (2, 0) theory of type
SU(N ) put on S1. In general, N = (2, 0) theory of type G has short multiplets containing a spacetime
symmetric traceless tensor that is in Vn2 of SO(5)R, for each n that gives a generator of invariant
polynomials of G. For G = SU(N ), n runs from 2 to N , thus giving (4.6) tensored with (3.8).
It would be interesting to perform similar computations for N = 2 theories for other G. For
example, when G = E6, we should have V0 V3 V4 V6 V7 V10, since the generators of invariant
polynomials of E6 have degrees 2, 5, 6, 8, 9, and 12. When G is non-simply laced, the UV completion
is N = (2, 0) theory of some simply laced type, with an outer-automorphism twist around S1. This
again predicts which Vn should appear among the one-instanton operators.
5. Quivers
In this section, we study the symmetry enhancement in the quiver gauge theory with SU gauge
groups and bifundamental hypermultiplets. In this note we do not aim at comprehensiveness; instead
we only treat the case where the effective number of flavors at each node SU(Ni ) is 2Ni and the
ChernSimons levels are all zero.
We use the by-now standard notation where stands for an SU(N1) flavor symmetry node and
an SU(N2) gauge symmetry node connected by a bifundamental hypermultiplet, etc. We also use
a special convention that two flavors of SU(1), , stand for . The rationale behind this
convention will be explained later in Sect. 5.3.
5.1. SU(2)2 theory
We denote the gauge groups as SU(2)1 SU(2)2. The hypermultiplets Q0, Q1, Q2 have flavor
symmetries SO(4)F0, SU(2)F1, SO(4)F2, respectively. The gauge group SU(2)1 effectively has N f = 4
flavors, and thus it has E4+1 = SO(10) flavor symmetry when SU(2)2 is not gauged. After
gauging, the remaining flavor symmetry is the commutant of SU(2)2, which is SU(4) SU(2). We can
summarize this enhancement pattern as
where the two white nodes on the left are SO(4)F0, the white node on the right is SU(2)F , and the
black node is the contribution from the instanton operator of SU(2)1.
The same argument can be applied to the SU(2)2 side, and we conclude that the full flavor symmetry
A B1 C
CB1 BA2 AB2 , (5.4)
where each symbol stands for a 2 2 block. The blocks A, A , and A are three SU(2) flavor
symmetries that can be seen in the Lagrangian; Bi comes from one-instanton operators of the SU(2)i gauge
group; and C comes from instanton operators that have instanton number one for both gauge groups
SU(2)1,2. Let us call these last ones (1, 1)-instanton operators. They transform as chiral spinors under
SO(4)F0,F2 and are neutral under SU(2)F .
Let us try to study the (1, 1)-instanton operators directly. The zero modes of gauginos of SU(2)1,2
give two copies of the broken current multiplets (3.2); those of hypermultiplets Q0 and Q2 give the
spinors of SO(4)F0 and SO(4)F2.
Finally, the hypermultiplet Q1 couples to one-instanton configurations of both SU(2)1,2. Therefore
this is effectively a triplet coupled to an SU(2) one-instanton configuration. It is also a doublet of
SU(2)F . Therefore they give rise to the states
where a, b = 1, 2 is now the index of the doublet of SU(2)F .
We therefore need to take the tensor product of two copies of the broken current multiplet (3.2),
the spinors of SO(4)F0,F2, and the multiplet (5.5). We then need to impose the Z2 projections for
SU(2)1,2.
At present we do not know enough about the behavior of the tensor product of the supersymmetry
multiplets in 5D. Instead, we learn the following by using the knowledge of the flavor symmetry
properties of (1, 1)-instanton operators deduced from the block decomposition (5.4): the tensor product
of two copies of the broken current multiplet (3.2) and the multiplet (5.5) contains again a unique
broken current multiplet. Furthermore, it is SU(2)F neutral, and therefore it comes from the factor
J+ in (5.5).
5.2. SU(N1) SU(N2) theory
We assume N1 > 2, N2 > 2, N0 + N2 = 2N1, and N1 + N3 = 2N2. We also set both the Chern
Simons levels to zero. Let us denote by U(1)B1 and U(1)B2 the baryonic flavor symmetries that
assign charge 1 to a field in the fundamental of SU(N1) and SU(N2), respectively. Let us also denote
by U(1)I1 and U(1)I2 the instanton number charge of SU(N1) and SU(N2), respectively.
We already saw that the combinations I1 := I1 B1/N1 and I2 := I2 B2/N2 are each
enhanced to an SU(2), giving SU(2)4 flavor symmetry. Let us show that I1+ and I2+ combine to
form an SU(3)+ and similarly that I1 and I2 combine to form an SU(3).
To see this, we need to analyze (1, 1)-instanton operators in this theory. The gauge group SU(Ni )
is broken to U(1)i SU(Ni 2). The gaugino zero modes can be analyzed as before. The
hypermultiplets Q0 and Q2 give doublets of SU(2) one-instanton configuration; the hypermultiplet Q1
similarly gives a lot of doublets and just one triplet of SU(2) one-instanton configuration.
Then, we need to find states that are neutral under the unbroken gauge group from the tensor
product of the following contributions:
1. From fields that are doublets of the SU(2) one-instanton configuration, we have
1a. contributions (4.2) from gauginos of SU(N1) and SU(N2),
1b. and contributions (4.3) from Q0, Q1, and Q2.
2. From fields that are triplets of the SU(2) one-instanton configuration, we have
2a. a contribution (5.5) from Q1,
2b. and two copies of the broken current multiplet (3.2) from SU(N1,2).
From the contributions 1, we find two states that are neutral under the unbroken gauge group
U(1)1 SU(N1 2) U(1)2 SU(N2 2), by tensoring the ground state or the top state of (4.3)
from the contributions 1a by the ground state or the top state of (4.3) from the contributions 1b.
From the contributions 2a, we note that the only U(1)1- and U(1)2-neutral state is the component
J in (5.5). Tensoring with the contributions 2b, we find a broken current multiplet, as we found at
the end of the last subsection.
In total, we find at least two gauge-invariant broken current multiplets. The charges under the
Lagrangian flavor symmetries can be easily found: both are neutral under SU(N0) and SU(N3), and
the charges under U(1)Q0, U(1)Q1, U(1)Q2 are
Therefore we have found two broken current multiplets with charges under (I1+, I2+; I1, I2)
given by (1, 1; 0, 0) and (0, 0; 1, 1), respectively. We already know that one-instanton operators of
SU(N1) give broken current multiplets with charges (1, 0; 0, 0) and (0, 0; 1, 0), and similarly those of
SU(N2) give multiplets with charges (0, 1; 0, 0) and (0, 0; 0, 1). Therefore we see that the instanton
number currents I1+, I2+ combine to form SU(3)+, and the currents I1 and I2 combine to form
SU(3). The total flavor symmetry is therefore at least
SU(3)+ SU(3) SU(N0) SU(N3) U(1).
The last U(1) is absent when N0 or N3 is zero.
5.3. Some special two-node quivers
We need to analyze separately the cases when one of the gauge groups is SU(2) or SU(1). As
already mentioned, we use the convention where two flavors of SU(1)
We also apply the same convention where the SU(2) on the left is gauged.
From a purely 5D field theory point of view, this is really just a convention, but it is useful because
SU(1) with two flavors shows an enhanced symmetry of SU(2) SU(2) SU(2), just as a special
case of SU(N ) with 2N flavors with the symmetry enhancement SU(2) SU(2) SU(2N ).
From a string/M theory point of view, when SU(1) with two flavors is engineered, say, in the
brane web construction, one in fact finds additional hypermultiplets coming from point-like SU(1)
instantons that naturally give rise to the setup (5.10). This is another rationale for our convention.
Now, let us couple this SU(1) with two flavors to an SU(2) gauge group to form a two-node
quiver:
Using our convention, this is just SU(2) with five flavors that show an enhancement to E6. This
contains SU(3)+ SU(3) SU(3), showing the general pattern that we found in (5.8).
We already treated
and saw that the symmetry is SU(2) SU(6) SU(2). As SU(6) SU(3)+ SU(3) U(1), it
again shows the general pattern (5.8).
Finally, let us consider
The SU(2) theory before coupling to SU(4) has an enhanced symmetry SO(10). After coupling to
SU(4) the remaining part is SU(2)+ SU(2), and they are enhanced by the dynamical SU(4) to
SU(3)+ SU(3), again following the general pattern (5.8).
5.4. Multi-node quivers
After our preparation on the two-node quivers, it is easy to analyze general multi-node quivers, again
with the restriction that each SU(N ) node has effectively 2N flavors and zero ChernSimons terms.
Consider as an example the quiver
Each SU(Ni ) node with Ni = 4, 3, 2, 1 shows an enhancement of the linear combination of the
instanton current and the baryonic current, Ii = Ii Bi /Ni to SU(2)i. For each neighboring pair
of nodes SU(Ni ) SU(N j ), SU(2)i and SU(2) j enhance to form SU(3). Therefore, in total,
we should have SU(5)+ SU(5) from the enhancement of the instanton number symmetry and
the baryonic symmetry. Combined with the original flavor symmetry SU(5) of the leftmost node, we
have
SU(5)+ SU(5) SU(5)
as the enhanced symmetry. We can easily generalize this analysis to an analogous linear quiver with
the gauge group SU(N 1) SU(N 2) SU(2) SU(1), with bifundamentals between
the neighboring gauge nodes and additional N fundamentals for SU(N 1); we see the symmetry
SU(N )+ SU(N ) SU(N ). The 5D SCFT is called the 5D TN theory, and this linear quiver
presentation was recently studied in Refs. [17,20,25,26].
As another example, consider the following quiver:
We have gauge groups SU(Ni ) with i = 1, 2, . . . , 9, with Ni = ki m. Let us define Ii = Ii Bi /Ni .
Then, applying exactly the same argument as above, we see that the currents Ii+ combine to form
(E8)+ and the currents Ii enhance to (E8). The total flavor symmetry is not quite their product,
however. The Cartan generator corresponding to a pure KK momentum of (E8)+ is
Bi does not act on the hypermultiplets and therefore is trivial. Thus we have
meaning that the total flavor symmetry is
ki Ii+ =
1
ki Ii + m
ki Ii+ =
ki Ii,
showing that the possible UV completion of this gauge theory is a 6D SCFT with flavor symmetry
E8 E8 on S1. This is as expected: m M5-branes on the ALE singularity of type E8 gives a 6D
N = (1, 0) SCFT in the infrared, with E8 E8 flavor symmetry. Compactifying it on S1 and
reducing it to type IIA, we have m D4-branes probing the ALE singularity of type E8. Using the standard
technique [46], we find the quiver theory given above.
The general statement is now clear. Take a 5D quiver gauge theory, with each SU(N ) gauge node
having effectively 2N flavors. If the quiver is a finite simply laced Dynkin diagram of type G, the
instanton number currents enhance to G G; if the quiver is an affine simply laced Dynkin diagram
of type G, the instanton number currents enhance to G G.
6. Conclusions
In this paper we have analyzed the one-instanton operators of 5D gauge theories with SU(N ) gauge
groups with hypermultiplets in the fundamental, adjoint, or bifundamental representations. We saw
that a simple exercise in the treatment of fermionic zero modes gives rise to the expected patterns of
symmetry enhancements.
There are many areas to be further explored. One is to extend our analysis to include SU(N ) gauge
theories with other matter representations, such as antisymmetric or symmetric two-index tensor
representations, and to consider other gauge groups, both classical and exceptional. There should
not be any essential difficulty in performing this generalization, since a one-instanton configuration
in any group G is always just an SU(2) one-instanton configuration embedded into G. Our analysis
of the SU quiver theory is by no means exhaustive, and it would be interesting to consider more
general cases.
It might be interesting to study instanton operators with higher instanton numbers. This will be
significantly harder, however, since the instanton moduli space is much more complicated.
Presumably, we will need to use the localization etc. to analyze it, and the method would become equivalent
to what has already been done in the literature in the study of the superconformal index of the 5D
SCFTs.
Another direction is to study in more detail the structure of the supermultiplets formed by operators
in non-conformal 5D supersymmetric theories. In this paper we relied on some heuristics based on the
known supermultiplet structures of superconformal theories. The gauge theories in the infrared are,
however, non-conformal, and we should analyze them as they deserve. For example, in our analysis
of SU(N1) SU(N2) theory, we could not directly analyze the tensor product decomposition of the
two copies of (3.2) and the contribution (5.5); instead we needed to import the knowledge gained by
the analysis of the special case SU(2)2. This is not an ideal situation. With a proper understanding
of the supermultiplet structures of operators in non-conformal theories, we would be able to analyze
this tensor product directly.
We also assumed throughout this paper that we only have to consider fermionic zero modes around
the one-instanton configuration, and that the states with excited non-zero modes do not give
broken current supermultiplets. This is at least plausible, since non-zero modes would likely produce
descendant operators, but this is not at all a rigorous argument. This needs to be better investigated.
Finally, we assumed in this paper that the gauge theory that we analyze is a mass deformation of a
UV fixed point, either a 5D one or a 6D one compactified on S1, and then studied what would be the
enhanced symmetry in the ultraviolet. It would be desirable to understand the criterion to tell which
5D gauge theory has a UV completion.
The author would like to come back to these questions in the future, but he will not have time in
the next few months due to various duties in the university. He hopes that some of the readers get
interested and make great progress in the meantime.
Y.T. thanks the Yukawa Institute at Kyoto University, for inviting him to give a series of lectures in February
2015. He, without much thought, promised to give an introduction to supersymmetric field theories in five
and six dimensions. It was after much thought about how to organize the content that he came up with the
arguments presented in this manuscript. He also thanks Naoki Kiryu, Tatsuma Nishioka, Kantaro Ohmori, and
Masato Taki for discussions. He is grateful to Kazuya Yonekura and Gabi Zafrir for pointing out an error in the
first version of this manuscript. The work is supported in part by a JSPS Grant-in-Aid for Scientific Research
No. 25870159, and in part by the WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo.
Open Access funding: SCOAP3.
[1] N. Seiberg , Phys. Lett . B 388 , 753 ( 1996 ) [arXiv:9608111 [hep-th]] [Search inSPIRE] .
[2] D. R. Morrison and N. Seiberg , Nucl. Phys .B 483 , 229 ( 1997 ) [arXiv:9609070 [hep-th]] [Search inSPIRE] .
[3] M. R. Douglas , S. H. Katz , and C. Vafa , Nucl. Phys . B 497 , 155 ( 1997 ) [arXiv:9609071 [hep-th]] [Search inSPIRE] .
[4] K. A. Intriligator , D. R. Morrison , and N. Seiberg , Nucl. Phys . B 497 , 56 ( 1997 ) [arXiv:9702198 [hep-th]] [Search inSPIRE] .
[5] O. J. Ganor , D. R. Morrison , and N. Seiberg , Nucl. Phys . B 487 , 93 ( 1997 ) [arXiv:9610251 [hep-th]] [Search inSPIRE] .
[6] E. Witten , [arXiv: 9507121 [hep-th]] [Search inSPIRE].
[7] A. Strominger , Phys. Lett . B 383 , 44 ( 1996 ) [arXiv:9512059 [hep-th]] [Search inSPIRE] .
[8] O. Aharony , and A. Hanany , Nucl. Phys . B 504 , 239 ( 1997 ) [arXiv:9704170 [hep-th]] [Search inSPIRE] .
[9] O. Aharony , A. Hanany , and B. Kol , J. High Energy Phys . 01 , 002 ( 1998 ) [arXiv:9710116 [hep-th]] [Search inSPIRE] .
[10] O. DeWolfe , A. Hanany , A. Iqbal , and E. Katz , J. High Energy Phys . 03 , 006 ( 1999 ) [arXiv:9902179 [hep-th]] [Search inSPIRE] .
[11] H.-C. Kim , S.-S. Kim , and K. Lee , J. High Energy Phys . 1210 , 142 ( 2012 ) [arXiv:1206.6781 [hep-th]] [Search inSPIRE] .
[12] A. Iqbal and C. Vafa , Phys. Rev. D 90 , 105031 ( 2014 ) [arXiv:1210.3605 [hep-th]] [Search inSPIRE] .
[13] D. Bashkirov , [arXiv:1211.4886 [hep-th]] [Search inSPIRE].
[14] O. Bergman , D. Rodrguez-Gomez, and G. Zafrir , J. High Energy Phys . 1308 , 081 ( 2013 ) [arXiv:1305.6870 [hep-th]] [Search inSPIRE] .
[15] O. Bergman , D. Rodrguez-Gomez, and G. Zafrir , J. High Energy Phys . 1401 , 079 ( 2014 ) [arXiv:1310.2150 [hep-th]] [Search inSPIRE] .
[16] L. Bao , V. Mitev , E. Pomoni , M. Taki , and F. Yagi , J. High Energy Phys . 1401 , 175 ( 2014 ) [arXiv:1310.3841 [hep-th]] [Search inSPIRE] .
[17] H. Hayashi , H.-C. Kim , and T. Nishinaka , J. High Energy Phys . 1406 , 014 ( 2014 ) [arXiv:1310.3854 [hep-th]] [Search inSPIRE] .
[18] M. Taki , [arXiv:1310.7509 [hep-th]] [Search inSPIRE].
[19] O. Bergman , D. Rodrguez-Gomez, and G. Zafrir , J. High Energy Phys . 1403 , 112 ( 2014 ) [arXiv:1311.4199 [hep-th]] [Search inSPIRE] .
[20] M. Aganagic , N. Haouzi , and S. Shakirov , [arXiv:1403.3657 [hep-th]] [Search inSPIRE].
[21] M. Taki , [arXiv:1401.7200 [hep-th]] [Search inSPIRE].
[22] C. Hwang , J. Kim , S. Kim , and J. Park , [arXiv:1406.6793 [hep-th]] [Search inSPIRE].
[23] G. Zafrir , [arXiv:1408.4040 [hep-th]] [Search inSPIRE].
[24] H. Hayashi and G. Zoccarato , [arXiv:1409.0571 [hep-th]] [Search inSPIRE].
[25] O. Bergman and G. Zafrir , [arXiv:1410.2806 [hep-th]] [Search inSPIRE].
[26] H. Hayashi , Y. Tachikawa , and K. Yonekura , [arXiv:1410.6868 [hep-th]] [Search inSPIRE].
[27] V. Mitev , E. Pomoni , M. Taki , and F. Yagi , [arXiv:1411.2450 [hep-th]] [Search inSPIRE].
[28] N. Lambert , C. Papageorgakis , and M. Schmidt-Sommerfeld , [arXiv:1412.2789 [hep-th]] [Search inSPIRE].
[29] D. Rodriguez -Gomez and J . Schmude, [arXiv:1501.00927 [hep-th]] [Search inSPIRE].
[30] D. Gaiotto and E. Witten , Adv. Theor. Math. Phys. 13 , 721 ( 2009 ) [arXiv:0807.3720 [hep-th]] [Search inSPIRE] .
[31] J. Bhattacharya , S. Bhattacharyya , S. Minwalla , and S. Raju , J. High Energy Phys . 0802 , 064 ( 2008 ) [arXiv:0801.1435 [hep-th]] [Search inSPIRE] .
[32] S. Terashima , Phys. Rev. D 89 , 125001 ( 2014 ) [arXiv:1207.2163 [hep-th]] [Search inSPIRE] .
[33] Y. Imamura and H. Matsuno , J. High Energy Phys . 1407 , 055 ( 2014 ) [arXiv:1404.0210 [hep-th]] [Search inSPIRE] .
[34] J. Kim , S. Kim , K. Lee , and J. Park , J. High Energy Phys . 1408 , 167 ( 2014 ) [arXiv:1405.2488 [hep-th]] [Search inSPIRE] .
[35] N. Dorey , T. J. Hollowood , V. V. Khoze , and M. P. Mattis , Phys. Rept. 371 , 231 ( 2002 ) [arXiv:0206063 [hep-th]] [Search inSPIRE] .
[36] C. W. Bernard , N. H. Christ , A. H. Guth , and E. J. Weinberg , Phys. Rev. D 16 , 2967 ( 1977 ).
[37] A. I. Vainshtein , V. I. Zakharov , V. A. Novikov , and M. A. Shifman , Sov. Phys. Usp . 25 , 195 ( 1982 ).
[38] M. Bianchi , M. B. Green , S. Kovacs , and G. Rossi , J. High Energy Phys . 9808 , 013 ( 1998 ) [arXiv:9807033 [hep-th]] [Search inSPIRE] .
[39] N. Dorey , V. V. Khoze , M. P. Mattis , and S. Vandoren , Phys. Lett . B 442 , 145 ( 1998 ) [arXiv:9808157 [hep-th]] [Search inSPIRE] .
[40] N. Dorey , T. J. Hollowood , V. V. Khoze , M. P. Mattis , and S. Vandoren , J. High Energy Phys . 9906 , 023 ( 1999 ) [arXiv:9810243 [hep-th]] [Search inSPIRE] .
[41] N. Dorey , T. J. Hollowood , V. V. Khoze , M. P. Mattis , and S. Vandoren , Nucl. Phys . B 552 , 88 ( 1999 ) [arXiv:9901128 [hep-th]] [Search inSPIRE] .
[42] C. N. Yang , J. Math. Phys. 19 , 320 ( 1978 ).
[43] Y. Tachikawa , J. High Energy Phys . 1111 , 123 ( 2011 ) [arXiv:1110.0531 [hep-th]] [Search inSPIRE] .
[44] M. R. Douglas , J. High Energy Phys . 1102 , 011 ( 2011 ) [arXiv:1012.2880 [hep-th]] [Search inSPIRE] .
[45] N. Lambert , C. Papageorgakis , and M. Schmidt-Sommerfeld , J. High Energy Phys . 1101 , 083 ( 2011 ) [arXiv:1012.2882 [hep-th]] [Search inSPIRE] .
[46] M. R. Douglas and G. W. Moore , [arXiv:9603167 [hep-th]] [Search inSPIRE] .